# Proof of logarithmic properties.

• pollytree
In summary, the two log properties are that ln(b1/n)=(1/n)ln(b) for b>0, and that ln(ar)=rln(a) for any r in Q and a>0.
pollytree

## Homework Statement

There are two log properties that I have to prove:
1) Explain why ln(b1/n)=(1/n)ln(b) for b>0, set b=an

2) Explain why ln(ar)=rln(a) for any r in Q and a>0, ie r is rational.

ln(an)=nln(a)

## The Attempt at a Solution

In a previous question I have already proved that ln(an)=nln(a), where n is a natural number. What I'm unsure about, is how is this any different? For 1), I'm not sure why you set b=an? Wouldn't you get ln((an)1/n) = ln(a)? I'm not sure how this helps me find the solution.

Similarly for 2), I'm unsure how it is any different to proving that ln(an)=nln(a).

Any help would be great!

The obvious difference is that you proved a result $$m ln(a)=ln(a^m)$$ when m is a natural number, and now they want you to do it when m is not a natural number. The proof that you used for this will not work when m is not a natural number (unless you did something really clever) because at some point you assume that it was, probably in order to write am as a*a*a...*a a multiplied together m times (which doesn't make sense if m=1/2 for example)

If b=an, what does ln(b) and ln(b1/n) become? Use it to do something cool

Office_Shredder said:
The obvious difference is that you proved a result $$m ln(a)=ln(a^m)$$ when m is a natural number, and now they want you to do it when m is not a natural number. The proof that you used for this will not work when m is not a natural number (unless you did something really clever) because at some point you assume that it was, probably in order to write am as a*a*a...*a a multiplied together m times (which doesn't make sense if m=1/2 for example)

If b=an, what does ln(b) and ln(b1/n) become? Use it to do something cool

Thanks for your help. I worked out part 1, but I'm still unsure on part 2. For proving the law for a natural number I did:
Let y=ln(a), for a>0,
ey=a, and if we raise each side to the power n we get:
(ey)n=an
ln(ey*n)=ln(an)
y*n = ln(an), but y = ln(a)
n*ln(a) = ln(an)

So I'm still unsure how it's any different for any rational number?
Any help would be great :)

Use the definition of Q, namely that r =m/n, where m,n are in N.

Ah I get it now! Thanks :)

## 1. What is the definition of logarithm?

The logarithm of a number is the exponent to which another fixed value, called the base, must be raised to produce that number.

## 2. What are the basic properties of logarithms?

The basic properties of logarithms are the product rule, quotient rule, power rule, and change of base rule.

## 3. How do you prove the product rule for logarithms?

The product rule for logarithms states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. This can be proved by using the definition of logarithm and basic algebraic manipulations.

## 4. What is the quotient rule for logarithms?

The quotient rule for logarithms states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers. This can also be proved using the definition of logarithm and algebraic manipulations.

## 5. Why is it important to understand the properties of logarithms?

Understanding the properties of logarithms is important in many fields, such as mathematics, science, and engineering. These properties allow us to simplify complex logarithmic expressions and solve equations involving logarithms, making calculations and problem-solving more efficient and accurate.

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